a plane in space
A flat, endless sheet — a tabletop extended forever in every direction — is a plane. How do you describe one precisely with coordinates? The slick answer turns on a single perpendicular arrow. If you fix one point on the sheet and the direction sticking straight out of it, the whole sheet is determined: it is every point P such that the arrow from your fixed point to P lies flat in the sheet, i.e. is perpendicular to that sticking-out direction.
Let the plane pass through a point A with position vector a, and let n = (A, B, C) be a normal vector — a vector perpendicular to the plane. A point with position vector r lies on the plane exactly when the displacement r - a is perpendicular to n, that is n . (r - a) = 0. Multiplying out gives the scalar (point-normal) equation A(x - x_0) + B(y - y_0) + C(z - z_0) = 0, which rearranges to the general form Ax + By + Cz = D, where D = n . a. The coefficients (A, B, C) are precisely a normal direction, which is the quickest thing to read off any plane equation. To build a plane through three non-collinear points, form two edge vectors and take their cross product to get n.
Planes are the flat building blocks of solid geometry, and the normal vector is the key that unlocks angles and distances: the angle between two planes (the dihedral angle) is the angle between their normals; the distance from a point to a plane uses the normal; two planes are parallel when their normals are parallel and perpendicular when their normals are perpendicular. A common confusion: the SAME plane has many equations, since you may scale (A, B, C, D) by any nonzero constant and the plane is unchanged. What matters is the DIRECTION of the normal, not its length — so always compare normals up to scaling when testing whether two equations describe one plane.
The plane through A(1, 0, 2) with normal n = (3, -1, 2) is 3(x - 1) - 1(y - 0) + 2(z - 2) = 0, i.e. 3x - y + 2z = 7. Test the point (2, 1, 1): 3(2) - 1 + 2(1) = 6 - 1 + 2 = 7, so it lies on the plane. The point (0, 0, 0) gives 0, not 7, so the origin is off the plane.
A point plus a normal direction fixes a plane; the coefficients are the normal.
In space, the linear equation Ax + By + Cz = D is a PLANE, not a line — a frequent slip for students arriving from 2D, where Ax + By = C was a line. The dimension changes what one linear equation means.